Vectors in the Essence of Linear Algebra

To make further qualitative progression in CLASSICAL ELECTRODYNAMICS, first we have to build a few Mathematical background by visualization.. we need Mathematics because the whole phenomena of Ectrodynamics are governed by 4 Maxwell's (Maxwell Heaviside) Equation of Electromagnetism along with Lorentz Force Law by Hendrik Lorentz and Special Relativity by Einstein..Today we'll discuss about vectors and the various operations vectors in the framework of linear algebra..

                                James Clerk Maxwell

                                  Oliver Heaviside 

                                   Hendrik Lorentz

                                       Albert Einstein 

VECTORS:-

First thing which makes us think what the heck are the vectors.. By definition Vectors are points in space which has specific Magnitude and direction.. Let's think about this for a second..

In nature we can fully describe some property only by numbers.. Like, what is the distance between two Poles, or what is the temperature etc.. But, in this conventional sense we can't really describe some properties like the force velocity etc...

Think about this for a moment if I tell you a car is moving at the speed 60 kmph, you can mentally picture how fast is the car really going but you can't tell me in which direction the car is moving by given data.. For that I have to give you some additional data like the car is moving with 60 kmph speed at the northward direction.. Now you can mentally get where the car is going.. likewise if tell you I am pushing a brick with 5 Newton force, you can imagine how hard I am pushing the brick but again you can't tell in which direction I am pushing..Similarly I have to give additional information about the direction...

Now here comes the elegance Mathematics.. Mathematics takes our some initial notions and elevates our notion in another level by simply arranging our preliminary notions in order..

Anyway, Suppose I am the observer and I am in the centre of all actions(origin).. Now I can simply draw 2 lines intersecting each other orthogonally at the origin(if I am in a two dimensional world or in a plane )or 3 lines (if I am in a three dimensional world).. Now we can point any points by only 2 or 3 numbers depending on the numbers of my spatial dimensions this whole apparatus is called
CO-ORDINATE SYSTEMS and the three lines are called AXES..

Here in two dimensional plane if I am at the origin(O) and I say that a car is going to the point A,B,C we can simply say that for A the car will go 2 steps at right and 5 five steps in northward directions.. Similarly we can describe that the car will go 4 steps left and 3 step backwards(the Negative symbol implies that the car is going in opposite directions)... This specific set of numbers(2,5) , (-4,-3) are called coordinates of the specific points and the lines OA, OB, OC along with their directions are called Vectors.. The point is are also vectors if the points represent certain values and directions..To grasp the last statement think about a point in a stream, at that point the water current has some magnitude as well as some Directions.. so, we can interpret points in fluid flow vector quantities...

And a bunch of non-zero Vectors are called Vector Field..

So, we can see that every two dimensional vectors can be expressed by (2×1) matrix and three dimensional vectors can be expressed as a (3×1) matrix.. The first term of (3×1) or (2×1) is the numbers of rows and the second number is the number of Columns... and each row represents a spatial dimension.. Or we can express vectors in the form of addition in three spatial dimensions..

The first, second and third rows of the Following (3×1) matrix consecutively represents the magnitude of F in X,Y,Z directions..And in the second denotion Fₓ Fᵧ Fz are the magnitudes of F in X,Y,Z directions respectively and î, ĵ, k̂ represents the vectors with the value 1 in the following directions..
So, our expression of Electric Field like this

And Magnetic Field by this

we can see any Vector can be represented as the linear combination of three unit vectors along in three spatial dimensions..

LINEAR TRANSFORMATION :-

From our previous discussion we've seen that every observer has their own
co-ordinate system.. Mathematically they are all equally valid as ours..But for the two separate points of views an point in space can't be described as one single universal description here co-ordinates..

Take this as an example, Suppose the distance between your house and your school is 10 km and the distance between your friend's house and the school is 3 km.. If none is lying then we know that your friend house is located some distance apart from your house...

Same goes here this two Coordinate Systems based on different observer's points of view are also connected in some specific way..For describing the whole relation we have to keep records of only two or three unit vectors based on our coordinate system..

Now if the unit vector along the X axis ( î ) of the second observer lands on (X₁, Y₁,Z₁) of the first observer and the unit vector along Y axis ( ĵ ) lands on (X₂,Y₂,Z₂) and the Z axis (k̂) lands on (X₃,Y₃,Z₃) then we can express the whole Transformation in a (3×3) matrix..

where each column represents the landing coordinates of the observer's three unit vectors (î, ĵ, k̂)..

And from our understanding of Linear Algebra we can say the new vector can also be written as the sum of the scaled version of new unit vectors... So, the transformation of a vector look like this.. 

Here x,y,z are the coordinates or the components of the input vector..

DOT PRODUCT:-

Dot product of two vectors are often described as the scaler multiplication of two vectors.. But what does it actually represents beyond of the mathematical definition??

In general dot product if two vectors is denoted by a dot between [ • ] the vectors and the geometrical interpretation of the operation is to project one vector on the other and scale the second vector by the factor of the projection.. But true meaning of a dot product from a natural perspective is really to measure the effect of one vector on another..

Take this as an example, Suppose you want to move a boulder in positive X direction but you are applying force on the boulder with some angle with its desired direction of movement.. will it move as it should be if both the direction of the force and the direction of it's movement were same?? Simple and intuitive answer is No.. From our everyday experience we can also tell the the greater the value of the angle between these two directions the less the boulder will move...

Dot product simply measures this effect numerically...

The above picture will clear the concept uptown certain extent..

So Geometrically the projected value of the vector A, on the vector B is |A|cosθ... This symbol ( | | ) represents only the magnitude of Vector A..

Now From Linear Algebra's point of view this whole operation looks like a certain type of transformation, which takes a vector from 2D plane and scales it's effect on a single line..

The transformation looks like this..

In linear algebra, we can prove that the first the first transformation is same as the vector with the same components..
So the transformation of the vector with the components (X,Y,Z) is same as the multiplication of the two vectors..

CROSS PRODUCT:-  

In general cross product between two vector is identical to dot product in meaning.. Like dot product it measures the effect of vector on another vector.. but this time number of vectors is increased..

In natural if we have two vectors, the cross product numerically produces their effect on the third vector perpendicular to both given vectors..

Let's think about this for a moment.. what is the best way to measure the effect two
co-planer vectors?? The answer is to measure their area in phase space..I mean there is no better way to sum up all the effects of both vectors..

This example is not pretty intuitive.. But here is the example... From various experiments we have seen that a charged particle if moves through a strong magnetic field then the magnetic field lines emitted by the particle(Maxwell's Corkscrew rule) face an repulsion force at every point on the plane and slowly moves to the direction orthogonal to both external Magnetic field line and the direction of the motion of the particle.. so, here the interactions between two co-planer vectors result as a third vector..

In the frame work of Linear Algebra this whole event looks like a vector with the magnitude, area enclosed by the first two co-planer is some how projected on a third vector..and from our knowledge of dot product we can say that this is exactly the dot product..

Here the middle term represents the volume of the three dimensional quadrilateral shape( namely Parallelopiped).. hence the î, ĵ, k̂ are three variables so, the whole term represents various volumes enclosed by the three vectors , while the area between the two given vectors are constant...

From the context we can arrive that cross product is the measurement of two co-planer vectors on the third vector..

Written by:- Shravan Ghosh